Modulated Fronts And Spreading Speed For A Monostable Nonlocal Reaction-Diffusion Equation

More accurately the reaction diffusion equations with nonlocal delays have attracted more and more attention,and thus become an important field in the study of partial differential equations,since they can describe natural phenomena in physics,chemistry and biology.In this paper,we study the asymptotic propagation velocity of the modulated wave and the corresponding Cauchy problem of a kind of nonlocal reaction diffusion equation with integral terms.The main content is divided into three chapters.Firstly,by disscussing bifurcation,we prove that Turing bifurcation of the equation occurs at the equilibrium u = 1.Further,with the help of the central manifold theorem,we prove the existence of periodic steady state and give its concrete form.Secondly,by considering a special kernel function and using the amplitude equation,we obtain an approximation of the modulated wave.Then,by applying the Center manifold reduction theorem,we prove that the equation has a modulated wave connecting the periodic steady state to the equilibrium u = 1.Finally,we study the corresponding Cauchy problem,through a series of analysis and discussion,we give the uniform boundness of the solution and corresponding asymptotic propagation velocity.